how do you find this limit?

Question

how do you find the limit of a function with indeterminate form of infinity-infinity
like this one:
lim as x approaches 0 of (1/tan x) - (1/x)

Answer 1

1/tanx - 1/x=
(x-tanx)/(xtanx)=0/0
so use l'hopital's rule
(1-sec^2(x))/(tanx+xsec^2(x))=
(1-1)/(0+0)= 0/0
so use l'hopital again
(-2secxsecxtanx)/(sec^2(x)+
sec^2(x)+x(2secxsecxtanx))=
0/(2+0)=0

Answer 2

l'hopital's rule!!!